# Question 7128: Five Prefix Sums and Four Ordered Differences [Return to catalogue page 36](https://ocme.evemisslab.com/en/questions/page/36/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261011-advance-20261015-r05-sequence-foundations/#q-20261011-advance-20261015-r05-sequence-0078) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0013-07128 · q-20261011-advance-20261015-r05-sequence-0078 ## Original question and givens N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-2, -1, 0, 1, 2]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?). - G001 — N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-2, -1, 0, 1, 2]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) ## Step-by-step derivation S001 · impliesBefore N starts at0;ordered finite indices and repeated values retained After given recurrences differ from guessing an infinite rule from observations Conditions: none Sequences are ordered and may repeat.A finite observed prefix alone does not determine a unique infinite rule;when a recurrence is explicitly supplied,use that rule rather than inventing one. Step identifiers and dependenciesRule: finite_ordered_sequence_scope · Uses: G001 S002 · calculationBefore finite prefix sum through0 := sum(0, -2) After finite prefix sum through0 = -2 Conditions: none Add the current term to the preceding prefix sum in original order,including repeats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore finite prefix sum through1 := sum(-2, -1) After finite prefix sum through1 = -3 Conditions: none Add the current term to the preceding prefix sum in original order,including repeats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore finite prefix sum through2 := sum(-3, 0) After finite prefix sum through2 = -3 Conditions: none Add the current term to the preceding prefix sum in original order,including repeats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore finite prefix sum through3 := sum(-3, 1) After finite prefix sum through3 = -2 Conditions: none Add the current term to the preceding prefix sum in original order,including repeats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore finite prefix sum through4 := sum(-2, 2) After finite prefix sum through4 = 0 Conditions: none Add the current term to the preceding prefix sum in original order,including repeats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore finite adjacent delta0 := sub(-1, -2) After finite adjacent delta0 = 1 Conditions: none Take each ordered next-minus-previous difference without sorting or deduplication. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore finite adjacent delta1 := sub(0, -1) After finite adjacent delta1 = 1 Conditions: none Take each ordered next-minus-previous difference without sorting or deduplication. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore finite adjacent delta2 := sub(1, 0) After finite adjacent delta2 = 1 Conditions: none Take each ordered next-minus-previous difference without sorting or deduplication. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore finite adjacent delta3 := sub(2, 1) After finite adjacent delta3 = 1 Conditions: none Take each ordered next-minus-previous difference without sorting or deduplication. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore [-2, -1, 0, 1, 2] After all five values equal=No Conditions: none These prefix sums are finite accumulations of five terms,not an infinite series or limit.The constant-value verdict concerns only these five terms. Step identifiers and dependenciesRule: finite_prefix_not_infinite_sum · Uses: G001, S010 ## Reverse checks - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute finite adjacent delta3 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute finite adjacent delta2 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute finite adjacent delta1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute finite adjacent delta0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute finite prefix sum through4 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute finite prefix sum through3 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute finite prefix sum through2 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute finite prefix sum through1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute finite prefix sum through0 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([-2, -3, -3, -2, 0], [1, 1, 1, 1], No) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: e6f151a0b4549dbf5c79470109cd7ef97ffe11e1d7e94d55ee359b597cf52f98 - candidate record SHA-256: f7defc11bd2a8af73e5e7b396a9281de8db3a7c84d0696465633870bcc8b4655 - semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59 - question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261011-advance-20261015-r05-sequence-0078.json)